I'm interested in finding all entire functions $g:\mathbb{C}\to\mathbb{C}$ such that $g(z+1)=g(z)$ and $g(z+i)=e^{2\pi}g(z)$, for all $z\in\mathbb{C}$.
My initial plan was to make use of Liouville's Theorem to show that $g$ is constant, but I realized later that the presence of the $e^{2\pi}$ factor might actually mean that this may not be true. Also, the fact that $e^{-2\pi iz}$ is such a non-constant function meant that my strategy may not work at all. At this point, I'm kind of unsure as to how to proceed, so any help or hint will be greatly appreciated. Thanks in advance.